1

Supplemental Material for: Recombination Kinetics

in Organic-Inorganic Perovskites: Excitons, Free

Charge and Sub-Gap States

Samuel D. Stranks1, Victor M. Burlakov

2, Tomas Leijtens

1, James M. Ball

1, Alain Goriely

2,

Henry J. Snaith1*

1Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK;

2Mathematical Institute, OCCAM, Woodstock Road, University of Oxford, Oxford, OX2 6GG,

UK

*Corresonding author: h.snaith1@physics.ox.ac.uk

I. Materials and Methods

Sample preparation. The perovskite precursor mixture was synthesized and dissolved at 40

wt% in dimethylformamide (DMF) as described previously [1, 2]. Microscope slides were

washed sequentially with soap (2% Hellmanex in water), de-ionized water, isopropanol, acetone

and finally treated under oxygen plasma for 10 minutes to remove the last traces of organic

residues. The precursor solution was spin-coated at 2000 rpm for 60 sec in air, and the substrates

were subsequently heated at 100°C in an oven in air for 45 min to allow perovskite crystal

2

formation. The samples were then sealed with a ~100nm layer of poly(methylmethacrylate)

(PMMA) (10mg/ml, 1000rpm) to prevent prolonged exposure to moisture.

Devices were fabricated as per methods published previously [2]. In brief, the perovskite

precursors were spin-coated onto an anode consisting of fluorinated tin oxide (FTO) coated with

an n-type TiO2 compact layer, heated, coated with a top layer of the p-type hole-transporter

Spiro-OMeTAD, and completed with thermally-evaporated Au counter-electrodes.

Optical Spectroscopy. Steady-state absorption spectra were acquired with a Varian Cary 300

UV/Vis spectrophotometer using an integrating sphere to account for optical losses outside of the

active layer.

Steady-state photoluminescence (PL) spectra and quantum efficiency (PLQE) values were

determined using a 532nm CW laser excitation source (Suwtech LDC-800) to illuminate a

sample in an integrating sphere (Oriel Instruments 70682NS), and the laser scatter and PL

collected using a fiber-coupled detector (Ocean Optics USB 2000+). The spectral response of the

fiber-coupled detector setup was calibrated using a spectral irradiance standard (Oriel

Instruments 63358). Excitation intensities were adjusted using optical density filters and the

highest intensities were obtained using a 25cm focusing lens to reduce the spot size. PLQE

calculations were carried out using established techniques [3].

Time-resolved PL decays were acquired using a time-correlated single photon counting

(TCSPC) setup (FluoTime 300, PicoQuant GmbH). Temperature-dependent measurements were

carried out in vacuo using an Oxford Instruments OptistatDN cryostat with a specialized fitting

for the TCSPC setup. Samples were photoexcited using a 507nm laser head (LDH-P-C-510,

PicoQuant GmbH) with pulse duration of 117ps, fluences of ~0.03 – 3 μJ/cm2/pulse, and a

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repetition rate of 300kHz. Samples were illuminated with the pulsed light until emission

stabilised and steady-state was reached. Low temperature PLQE values were acquired by scaling

spectra from room temperature values that had been acquired using an integrating sphere. The

excitation repetition rate was high (80MHz) to ensure quasi-steady-state illumination conditions.

II. Equilibrium concentrations of free carriers and excitons in the presence of doping

Consider entropy of the system containing the concentrations ne and nh of free electrons and

holes, respectively, and the concentration nx of excitons with the binding energy bE

! ! !ln ln ln

! ! ! ! ! !e h x

B B B

e e e h h h x x x

M M MS k k k

M n n M n n M n n

(1)

where Mi are the concentrations of available states for i-th particles. According to Stirling’s

formula ln ! lnn n n n we may write

!

ln ln ln ln! !

ln

iB B i i i i i i i i

i i

iB i i

i

Mk k M M n n M n M n

M n n

nk n n

M

(2)

where we took into account that i in M . Rewriting Eq. (1) we then obtain

ln ln lne h xB e h x e h x

e h x

n n nS k n n n n n n

M M M

(3)

The total free energy G can be written as

4

ln ln

ln

e g x g b B x x x B e e e

B h h h B e h x

G n E n E E k T n n v k T n n v

k T n n v k T n n n

(4)

where 1 3,

2i i i i

i B

hM v

m k T

is the volume occupied by each quasiparticle with i

being the thermal wavelength. Suppose there are N photo-generated free electrons per cubic

centimetre some of which form excitons, i.e. e xn N n . In the presence of doped holes

concentration nT the total hole concentration is

h T e x Tn n n N n n (5)

Substituting this into Eq. (4) we obtain

ln

ln ln

2

x g x g b B x x x

B x x e B x T x T h

B x T

G N n E n E E k T n n v

k T N n N n v k T N n n N n n v

k T N n n

(6)

Thermal equilibrium corresponds to minimum of G ( / 0xdG dn ) resulting in the equation for

nx

ln 0b x x

B x T x h e

E n v

k T N n n N n v v

(7)

It is convenient to rewrite it as an equation for nh

0, expx bh T h T h

h e B

v En n n A N n n A

v v k T

(8)

5

This is a generalization of the Saha equation [4-6] for the case of p-doped semiconductors. For

n-doped semiconductors, the corresponding equation is obtained by swapping e- and h-

subscripts in Eq. (8). Note that 16 38 10A cm at T=300 K and

16 31 10A cm at T=190 K.

Solution to Eq. (8) is

21

42 2

T

h T

A nn A n A N

(9)

One can see that this solution depends on the doping concentration nT. In the case that doping

is due to the presence of electron sub-gap states, nT depends on the initial concentration N(0). To

obtain an explicit expression for nh as a function of N consider the steady state balance for nT

assuming that nT is a very slow variable that is dependent only on average concentration

0

0 0

1t

e en t n t dtt

of electrons photo-generated during the PL recording time 6

0 10t s

2 0Tpop T T e dep T T e

dnR N n n t R n n n t

dt (10)

where andpop depR R are the rates for the trap population and depopulation, respectively. Taking

the expression for en t from the section below (Eqs(15) and (17)) we evaluate average electron

concentration

0

0 0 0 00 0

01 1 1, ln 1

t

e e e T

T T

ANn t n t dt n t dt K A n K

t t t N A N

(11)

6

where 0 eh xAR R accounts for the total rate of electronic decay not involving sub-gap states.

For simplicity we replaced nT with NT in the (slowly-varying) logarithmic function. Substitution

of this into Eq. (10) gives

2

2 2

11 0

1 10, 4

2 2

,1 1

1 1

pop pop T pop pop

T T T

dep dep dep dep

T T T T T

pop

T

dep pop

pop pop

dep

dep dep

R R N R A RN A n A n

R R R K R

n n N n N

RA A N

R R A

R RR

K R K R

(12)

Together with Eqs (5) and (9) this allows us to obtain the concentrations of free carriers and

excitons as a function of initial charge concentration N(0) at different temperatures. The

equations without the presence of sub-gap states (no doping, NT = 0 hence nT=0) reduce back to

the Saha equation [4-6].

The results are presented in Fig. S1 below, where the dashed lines correspond to the case

without sub-gap states (no doping), which is analogous to the results obtained by D’Innocenzo et

al. using the Saha equation [6]. Our results deviate from those of the undoped material, meaning

that even low-level doping can affect the exciton concentration. Nevertheless, we also conclude

that the dominant species are free charges (≤10% excitons) under operating conditions in a

perovskite device at 300K (Nmax~1015

cm-3

). Only at higher charge densities or low temperatures

do we see larger fractions of excitons.

7

Figure S1. Fraction of excitons as a function of the photo-generated charge concentration N at

two temperatures. The model parameters for T=300 K are: 162.5 10TN cm

3,

7

0 1.4 10 s-1

,

102 10popR cm3s

-1, 128 10depR cm

3s

-1, and for T=190 K:

162.5 10TN cm-3

,

6

0 6.0 10 s-1

, 102 10popR cm3s

-1, 122 10depR cm

3s

-1. The dotted lines show the case

without sub-gap states (no doping).

III. Kinetics of free charge carriers and excitons

Suppose we have free carriers excited by laser pulse high up the conduction band (CB). They

quickly decay down to the bottom of CB due to electron-phonon interaction on the time scale ~1

ps (10-12

s). Electronic traps are filled also very quickly and we assume that they stay fully filled

on the time scale of the PL decay. Photo-generated electrons may form excitons or recombine

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straight away. As the hole concentration is always related to electron concentration as

h e Tn n n , we may write equations for electrons and excitons only

ef e h eh e h d x pop e T T

xf e h d x x x

dnR n n R n n R n R n N n

dt

dnR n n R n R n

dt

(13)

where fR is exciton formation rate, dR is exciton dissociation rate, ehR is free electron-hole

recombination rate, xR is exciton dissipative decay rate (not involving dissociation). The sum of

the two equations (13) with the condition that x e hAn n n (see Eq. (8)) gives

0e x

e h pop e T T

d n nn n R n N n

dt A

(14)

To simplify further these equations we approximate the solution Eqs (9) assuming that for

most of the time 2

4 TAN A n and therefore

, ,Th T x T h e h T

T T T

N nA N A Nn n n N n n n n n

A n A n A n

(15)

Then Eq. (14) can be rewritten as

0

0

,pop T TT

T T

R N nA ndx Nx x x

dt A n A A n

(16)

The solution to this equation is

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1 0 0

1 0

1

0

exp, ,

1 exp

pop T TT

T T T

pop T TT

AR N nC x t nNx

A n A n A nC x t

R N nnC

A

(17)

i. Photoluminescence

The time-dependent normalized PL intensity is

1 0 1 0

0 0 1 0 1 0

0 0 0 0

exp exp1

1 exp 1 exp

eh x e h

eh x e h

T

T

I t I t n t n t

I I n n

AC x t AC x tn

n Ax Ax C x t C x t

(18)

where Tn is given by Eq. (12).

ii. Photoluminescence Quantum Efficiency (PLQE)

For the pulsed excitation regime the PLQE can be defined as

2_ 1

0 10

0 01ln 1

0 0

rad rec pop T T

eh

T T

R R N nN NA CPLQE I t dt

N N A n C A n

(19)

where _rad recR is a parameter corresponding to the radiative component of exciton (or free

electron-hole) recombination.

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IV. Fitting procedure

We fit the model to our experimental data as follows. There are four main fitting parameters,

namely NT, γ0, Rpop and Rdep. The value of trap concentration NT can be estimated from the fact

that at the pulsed excitation level 150 10N cm-3

the PL decay (see Fig. S3) is nearly mono-

exponential suggesting that in Eq. (18) x0<<C1, or N<< Tn ~NT. In contrast, at 170 10N cm-3

the PL decay is highly non-monoexponential meaning that N>> Tn ~NT. Hence one may

conclude that a reasonable estimate for NT would be ~1016

cm-3

. The mono-exponential decay is

determined by γ (see Eqs (18)). The experimental value [2] of 1 is close to 300 ns, i.e.

6 13 10 s . As / / 0.1T T T Tn A n N A N the parameter 0 eh xAR R should have

the value ~107 s

-1.

The values of the trap population popR and depopulation depR rates are obtained from fitting

the shape of PLQE at room temperature (see Figure S2) and found to be 102 10popR cm3s

-1

and 128 10depR cm3s

-1. Using these values we performed fitting of the PL intensity decay (see

Figure S3) to rectify the parameter values, which then have been used to calculate the PLQE

again. This iteration procedure was repeated until satisfactory agreement for both PLQE and PL

intensity is achieved.

A similar approach was used to fit the data at T=250K and 190K, the results of which are

shown in Fig. S4 and S5, respectively. We also demonstrate in Fig. S6 for 190K that the total

trap concentration must change with temperature to allow good fits to the data at lower

temperature.

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Figure S2. PLQE as a function of photo-generated carrier concentration N. Experimental data

(symbols) versus the model (Eq. (19)) (solid line). The parameters are the same as in the caption

for Fig. S1. We also use_ 00.15rad recR cm

3s

-1, which is simply a scaling coefficient not used in

further calculations.

=1.4*10 7 s -1

T=300 K

Carrier concentration (cm -3

10 12

10 13

10 14

10 15

10 16

10 17

10 18 0.00

0.05

0.10

0.15

0.20

0.25

N T =2.5*10 16

cm -3 , R pop =2*10 -10

cm -3 s -1

R dep =8*10 -12

cm -3 s -1 , A*R eh +R

x

0.00

PLQE

)

12

Figure S3. PL intensity as a function of time after pulsed excitation at 300K. Experimental data

(noisy lines) versus the model (Eq. (18)). The model parameter values are the same as in Fig. S2.

Figure S4. PL intensity as a function of time after pulsed excitation at 250K. Experimental data

(noisy lines) versus the model (Eq. (18)). The model parameter values are also shown.

13

Figure S5. PL intensity as a function of time after pulsed excitation at 190K. Experimental data

(noisy lines) versus the model (Eq. (18)). The model parameter values are also shown.

Figure S6. PL intensity as a function of time at 190K, where good fits to the data in Fig. S5 are

obtained if the trap concentration is able to change from the room temperature value

(NT=0.9x1016

cm-3

, thick lines), and poor fits are obtained otherwise (NT=2.5x1016

cm-3

, dashed

lines).

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V. Steady state charge concentration at continuous excitation

To find N as a function of the system parameters (various rates and excitation flux I) we have

to solve the simultaneous equations

0

2

0

0

e x

e h pop e T T

Tpop T T e dep T T e

d n n In n R n N n

dt d A

dnR N n n R n n n

dt

(20)

where d is the film thickness. Solving these equations numerically as a function of I gives the

plot shown in Figure S7 below. The fraction of excitons to free charges is also very similar to

that obtained under the pulsed excitation regime, as shown in Figure S8.

Figure S7. Concentration of filled traps nT (solid symbols) and of photo-generated charges N

(open symbols) at 300 K (red) and 190 K (blue) as a function of incident photon flux density I.

The dashed line represents the incident absorbed flux for the perovskite under 1 sun illumination

(I~1.6x1017

/cm2/s), giving an upper bound on the room temperature open-circuit charge density

of N~8x1014

cm-3

. The sample thickness is 500nm.

15

Figure S8. Fraction of excitons as a function of the photo-generated charge concentration N at

two temperatures to compare the pulsed and steady state excitation regimes.

To calculate PLQE in the steady state regime we consider various radiative decay channels:

radiative decay of excitons and radiative decay of electrons and holes. For each case PLQE is

defined as a ratio of the decay rate through the channel to the total decay rate. Accordingly the

PLQEs (maximum because assuming Rex and Reh are purely radiative) are:

x xx

x x x x T eh T T pop

x T eh

eh

x x x x T eh T T pop

n RPLQE

n R N n N n n R N n R

N n n RPLQE

n R N n N n n R N n R

(21)

The corresponding PLQEs were calculated using the results for nT, N and nx (Fig S7 and S8)

and are shown in the main text. The values of Rx and Reh were calculated taking into account the

experimental saturation values (where all traps are filled) for PLQEx or PLQEeh (0.25 for 300 K,

0.45-0.7 for 250 K, and 0.95 for 190 K from Figure 3c in the main text).

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We compare the extracted values for the recombination rates (Reh and Rx) for the two cases in

Fig. S9. If electron-hole recombination is the primary radiative decay channel (Fig. S9a) then

with the decrease of T from 300 K down to 190 K, the value of Rx decreases ~30 times while Reh

increases 10 times. The large decrease in excitonic recombination rate and concomitant increase

in electron-hole recombination rate is required to compensate for the significant exciton fraction

at 190K (Fig. S8), species which are non-radiative in this case. However, these changes with

such a small temperature decrease (300 K down to 190 K) are much larger than observed in other

systems where electron-hole recombination is radiative [7, 8]. In particular, the 10-times increase

in the radiative decay rate may seem too high to be physically realistic [7, 8]. In contrast, if only

excitons decay radiatively then for the same decrease in temperature we find that Rx increases 2

times and Reh decreases ~5 times (see Fig. S9b), which is much more reasonable for this

temperature change. Therefore we deduce that emission may be preceded by formation of an

exciton, but we note that this does not exclude electron-hole recombination as a secondary

radiative pathway.

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Figure S9. Extracted electron-hole (Reh) and exciton (Rx) recombination rate constants for cases

where the only radiative channels are (a) electron-hole recombination (PLQEeh from Eq. 21) or

(b) excitonic recombination (PLQEx from Eq. 21).

VI. Dependence on Exciton Binding Energy

The results presented above and in the main text are obtained using an exciton binding energy of

50meV [6]. We now show that the model yields similar conclusions when using exciton binding

energies on the lower end of those reported in the literature for these materials, e.g. 15meV [9].

Figure S10 shows the fits to the PL decays. The extracted parameters are similar to those

obtained with an exciton binding energy of 50meV. Notably, the trap density does not change

significantly. Figure S11 shows that the filled trap and photogenerated charge densities are

insensitive to the binding energy over this range. However, as expected, the fraction of excitons

(Figure S12) does noticeably change.

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Figure S10. PL intensity as a function of time after pulsed excitation at 300K with an exciton

binding energy of 10meV. Experimental data (noisy lines) versus the model (Eq. (18)). The

model parameter values are also shown.

Figure S11. Concentration of filled traps nT (solid symbols) and of photo-generated charges N

(open symbols) at 300 K as a function of incident photon flux density I with two exciton binding

energies. The sample thickness is 500nm.

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Figure S12. Fraction of excitons as a function of the photo-generated charge concentration N

with two different exciton binding energies.

VII. Low temperature PLQE Measurements

The PL spectra of the thin perovskite films at varying temperatures through the range in which

the perovskite crystal remains in the tetragonal phase (155 – 320K) [6, 10, 11] are shown in

Figure S13, with an excitation fluence of ~2nJ/cm2/pulse and a high pulse repetition rate of

80MHz (i.e. quasi-steady state ~150mW/cm2 illumination). This corresponds to a charge density

in the perovskite film of ~2x1015

cm-3

(Fig. S7). The room temperature spectrum was integrated

and assigned the room temperature PLQE value from measurement in the integration sphere

(~8%). The other spectra (Figure S13) were also integrated and the PLQE values scaled

accordingly. A similar analysis was also carried out at different excitation fluences to obtain the

values presented in Figure 3c of the main text.

20

Figure S13. PL spectra as a function of temperature with pulsed 507-nm excitation at a

repetition rate of 80MHz to give quasi-steady state ~150mW/cm2 (~2x10

15 cm

-3 charge density)

illumination conditions.

VIII. References

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Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites, Science 338, 643

(2012).

[2] S. D. Stranks, G. E. Eperon, G. Grancini, C. Menelaou, M. J. Alcocer, T. Leijtens, L. M.

Herz, A. Petrozza, and H. J. Snaith, Electron-hole diffusion lengths exceeding 1 micrometer in an

organometal trihalide perovskite absorber, Science 342, 341 (2013).

[3] J. C. de Mello, H. F. Wittmann, and R. H. Friend, An improved experimental

determination of external photoluminescence quantum efficiency, Advanced Materials 9, 230

(1997).

[4] M. N. Saha, On a Physical Theory of Stellar Spectra, Proceedings of the Royal Society of

London. Series A 99, 135 (1921).

[5] J. Szczytko, L. Kappei, J. Berney, F. Morier-Genoud, M. T. Portella-Oberli, and B.

Deveaud, Determination of the Exciton Formation in Quantum Wells from Time-Resolved

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Lanzani, H. J. Snaith, and A. Petrozza, Excitons versus free charges in organo-lead tri-halide

perovskites, Nat Commun 5, 3586 (2014).

21

[7] Y. P. Varshni, Band-to-Band Radiative Recombination in Groups IV, VI, and III-V

Semiconductors (I), physica status solidi (b) 19, 459 (1967).

[8] H. Schlangenotto, H. Maeder, and W. Gerlach, Temperature dependence of the radiative

recombination coefficient in silicon, Physica Status Solidi (a) 21, 357 (1974).

[9] J. Even, L. Pedesseau, and C. Katan, Analysis of Multivalley and Multibandgap

Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid

Perovskites, The Journal of Physical Chemistry C 118, 11566 (2014).

[10] T. Ishihara, J. Takahashi, and T. Goto, Optical properties due to electronic transitions in

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